Limit Distributions of Eigenvalues for Random Block Toeplitz and Hankel Matrices

TL;DR

This paper investigates the eigenvalue distributions of random block Toeplitz and Hankel matrices, establishing their convergence to the semicircle law as block size increases, with explicit formulas for moments and trace representations.

Contribution

It provides the first proof of the almost sure convergence of eigenvalue distributions of these matrices to the semicircle law, including trace formulas and moment calculations.

Findings

Eigenvalue distributions converge to the semicircle law as block size increases.

Explicit trace formulas for block Toeplitz and Hankel matrices are derived.

Moments of the limit distributions are explicitly calculated.

Abstract

Block Toeplitz and Hankel matrices arise in many aspects of applications. In this paper, we will research the distributions of eigenvalues for some models and get the semicircle law. Firstly we will give trace formulae of block Toeplitz and Hankel matrix. Then we will prove that the almost sure limit $\gamma_{_T}^{(m)}$ $(\gamma_{_H}^{(m)})$ of eigenvalue distributions of random block Toeplitz (Hankel) matrices exist and give the moments of the limit distributions where $m$ is the order of the blocks. Then we will prove the existence of almost sure limit of eigenvalue distributions of random block Toeplitz and Hankel band matrices and give the moments of the limit distributions. Finally we will prove that $\gamma_{_T}^{(m)}$ $(\gamma_{_H}^{(m)})$ converges weakly to the semicircle law as $m\ra\iy$.